TY - JOUR

T1 - A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

AU - Kehle, Christoph

AU - Shlapentokh-Rothman, Yakov

N1 - Funding Information:
The authors would like express their gratitude to Mihalis Dafermos for many valuable discussions and helpful remarks. The authors also thank Igor Rodni-anski, Jonathan Luk, and Sung-Jin Oh for useful conversations. CK Acknowledges support from the EPSRC and thanks Princeton University for hosting him as a VSRC. YS acknowledges support from the NSF Postdoctoral Research Fellowship under Award No. 1502569.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - We develop a scattering theory for the linear wave equation □ g ψ= 0 on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies ω and ℓ. This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants Λ , there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior.

AB - We develop a scattering theory for the linear wave equation □ g ψ= 0 on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies ω and ℓ. This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants Λ , there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior.

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U2 - 10.1007/s00023-019-00760-z

DO - 10.1007/s00023-019-00760-z

M3 - Article

AN - SCOPUS:85061580457

VL - 20

SP - 1583

EP - 1650

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 5

ER -